Implementation of Statistic Process Control in a Painting Sector of a Automotive Manufacturer

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1 4 th International Conference on Industrial Engineering and Industrial Management IV Congreso de Ingeniería de Organización Donostia- an ebastián, etember 8 th - th Imlementation of tatistic Process Control in a Painting ector of a Automotive Manufacturer J. Cordeiro, J. G. Requeijo Dto. Deartamento de Engenharia Mecânica e Industrial, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, 89-56, Caarica, Portugal. UNIDEMI, Dto. de Engenharia Mecânica e Industrial, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, 89-56, Caarica, Portugal. Abstract The continuous imrovement on quality of roducts and rocesses is a constant concern at organizations, as a resonse to growing cometition and demands of the market. The imlementation of statistical techniques adjusted to different situations is one way to achieve this goal. The alication of traditional control charts requires, among other, that collected data are indeendent. However this is not always assured, reflecting a drastic increase of false alarms. This article resents a methodology for alication of traditional univariate control charts, based on residuals, when data exhibit significant autocorrelation. Imlementation took lace in the ainting rocess of an automotive comany. Keywords: tatistic Process Control; Traditional hewhart Control Charts; Autocorrelated Data; ARIMA Models. Introduction The seek for rocess imrovement in industrial organizations imlies a better understanding of the nature, interretation and modeling of variability. tatistical Process Control (PC) is a useful tool in the detection of unusual ways of variation, allowing acting on the same. The control charts, a technique for real-time monitoring of the rocess, is an aroriate instrument for this urose. The univariate control charts were introduced by Walter A. hewhart, and are currently adoted and imlemented in large-scale industrial rocesses, both discrete and continuous. One of the conditions for the alication of these techniques is the existence of data indeendence, not always found in an industrial environment. If data correlation is disregarded, erroneous conclusions can be taken and derail the identification of statistical control and deviations. According to Gilbert et al. (997), it is imerative to assess if the behavior of autocorrelated data is natural or indisutable to the rocess. To overcome such adversity, several aroaches have been develoed regarding the alication of univariate control charts to data coming from autocorrelated rocesses. These aroaches are divided into two distinct lines of investigation, a first aroach which imlies the determination of a mathematical model that best fit the autocorrelated data, and a second without of a mathematical model adjustment. The statistical control of rocesses with autocorrelated data has been aroached by several researchers, such as Alwan and Roberts (988), Montgomery and Mastrangelo (99), Ross and Harris (99), Maragah and Woodall (99), Wardell et al. (99), Vander Weil (996), Zhang (998), Wieringa (999), Woodall () and Jay et al. (8), among others. 565

2 This aer aims the investigation of the erformance and adequacy of univariate traditional control charts to data that exhibit autocorrelation and develos a methodology that allows the alication of most aroriate techniques, using the rocess modeling through ARIMA models. The study was erformed in a Portuguese automotive industry based on real data.. Traditional PC For a roer imlementation of traditional hewhart control charts, the samling must be carried out at aroriate time intervals and sufficient times, so that the collected observations can show the rocess behavior and maximize the chance of variation between samles; samles should be homogeneous, and it is exected that the units have been roduced consecutively and similarly; data, of the characteristic to be controlled, are considered indeendent and identically distributed according to a Normal distribution with mean and variance ; the distance between control limits and centre line of the several charts is 3 standard deviations of the distribution of samling statistic to be controlled, which corresonds to a significance level of.7%. There are several references, such as Quesenberry (997) and Pereira and Requeijo (8), which consider that the rocedure for construction of control charts can be distinguished in, at least, two hases. A first hase (Phase I), where the interest lies in concluding if the data from the ast come from a controlled rocess; a second hase (Phase II), where the control charts are used to monitor the rocess in real time. It is imortant to note that Phase II should only be initiated when Phase I of the rocess is under statistical control, and when the caability to roduce in accordance with the required technical secification is verified... Phase I In this hase, named as reliminary hase or retrosective, the uer control limit (UCL), lower control limit (LCL), center line (CL) and the rocess arameters are estimated. These estimations are made based on collected data, and on the equations resented in Table. Table. Control Limits of hewhart Control Charts for Phase I and estimators of rocess arameters Chart UCL CL LCL ˆ ˆ (individual observations) 3 MR d 3 MR d (mean) R (range) ( standard deviation) MR (moving range) A R or A3 A R or A3 D 4 R R R B 4 D 4 MR MR MR R D 3 B 3 4 MR D 3 The arameters (average and standard deviation) are estimated and subsequently the rocess caability is analyzed. Usually the caability indexes used for Normally distributed data are, C LE LIE 6 and C k min C k, C k, where C k LIE 3 and C k LE 3. Usually the rocess is considered caable, when C and I I d c d C k are simultaneously higher than a reference value k (k =.33 for bilateral technical secifications and k =.5 for unilateral technical secifications). As mentioned, it is assumed that the data of the characteristic to be controlled are indeendent and normally distributed. To study the normal distribution, it is suggested the alication of 566

3 the Kolmogorov-mirnov test and to study the indeendence of data the alication of Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF). After checking the stability of the rocess, if it does not ossess the caability to roduce at the technical secification required, corrective actions should be taken, leading to the collection of new data and reeat the Phase I of the PC... Phase II After the verification of stability, arameters estimation, and rocess caability analysis, the next hase is the monitoring, a rocedure commonly referred to as Phase II of PC. The values of the center line and control limits are estimated based on rocess arameters estimation undertaken in Phase I (equations resented in Table ), being similar to Phase I if no change of samle size exists. Table. Limits hewhart Control Charts for Phase II Chart UCL CL LCL (individual observations) 3 3 (mean) A A R (range) D d D (standard deviation) B 6 c 4 B 5 MR (moving range) D d D At this stage it is assumed that the rocess is statistically controlled and the rocess arameters are known, so when a secial cause is resent the reason for its occurrence should be investigated and corrective measures taken to roceed to its elimination. 3. PC for autocorrelated data As reviously mentioned, the use of hewhart control charts is based on the assumtion of statistical indeendence of the data. When this rincile is violated, there are two distinct aroaches; a first aroach which imlies the determination of a mathematical model that best fit the autocorrelated data, and a second without of a mathematical model adjustment. The first aroach involves the study of available data and verifies the tye of autocorrelation resent in the rocess, followed by its modeling. In this article we will follow this aroach, modeling the rocess using the ARIMA models. After the determination of residuals, or rediction errors, traditional control charts are alied to these variables, following the methodology roosed for the usual traditional PC referred to in section. For the second aroach, it is imortant to refer studies of Montgomery and Mastrangelo (99) and Montgomery and Mastrangelo (995). They roose the use of statistical EWMA (exonentially weighted moving average) for data from ositively autocorrelated rocesses, as well as the MCEWMA chart ((EWMA Moving Center Line), which allows simultaneously checking the rocess control state and monitoring its behavior. Zhang (998) resents the EWMAT chart (EWMA for stationary rocesses), which is nothing more than a change in the control limits of EWMA control chart through the autocorrelation function k. 3.. hewhart charts for autocorrelated data Modeling based on ARIMA models, lays a filter role that allows the elimination of the existing rocess autocorrelation, making residuals indeendent and normally distributed. With these residuals, hewhart control charts are designed for Phase I of PC. Later, when the rocess is under statistical control, forecasts will be carried to subsequent eriods and 567

4 calculated the rediction errors. These rediction errors reresent the set of data to be monitored in Phase II of PC hewhart charts for residuals control Like PC for indeendent data, the techniques to be imlemented in Phase I are the traditional hewhart charts alied to residuals (determined with the rocess modeling). In Table 3 the equations that allow the estimation of control limits and the standard deviation of the residuals,, are resented. Table 3. Control Limits for Phase I of control charts based on residuals and estimators Chart UCL CL LCL ˆ e ( residuals) 3 MR d e (mean) R (range) ( standard deviation) MR (moving range) 3 MR d A R or A 3 A R or A 3 D 4 R R R B 4 D 4 MR MR MR The distribution of residuals has an exected zero mean value and variance R d D 3 c B 3 4 MR d D 3. Based on the adjusted ARIMA model (AR (), MA (q) or ARMA (, q)), the location ( ) and disersion ( ) rocess arameters are estimated. These models will be described later on this aer. It is assumed that the residuals, concerning to the characteristic under study, are indeendent and Normally distributed. It is suggested that verification of the Normality of the residuals is checked using the Kolmogorov-mirnov test, and the indeendence by the ACF and PACF. When a secial cause of variation is found, the oint should not be eliminated but relaced by the exected value for that moment, setting back the ARIMA model, calculate the new residuals, and review the control charts. If there are many oints out of control, it is necessary to investigate the cause(s) that led to this situation and roceed to the necessary corrective actions. During the checking of the stability of the rocess, if it does not reveal caability to roduce according to technical secification, corrective actions should be taken. One of the objectives from Phase I of the PC is to estimate roerly the rocess arameters. When the quality characteristic shows significant autocorrelation, this estimation is erformed based on the arameters of the ARIMA model. The estimators of rocess mean and variance for the different models are given by the equations on Table 4. In the equations of Table 4 it is considered the variance of residuals, the arameter that determines the rocess average, j the order arameter j of AR or ARMA model, j the order arameter j MA or ARMA model, j the correlation coefficient of lag j and j auto-covariance lag j. The study of the rocess caability is erformed using the traditional caability indexes of rocess C and C, reviously defined. k 568

5 Table 4. Estimators of mean and variance of rocess, on the quality characteristic Model Mean Variance E Var AR j j j j j q MA E Var j, θ j ARMA E j j Var j j j q q The monitoring of future data, for autocorrelated rocesses, is erformed through the alication of hewhart control charts based on the rediction errors. The distribution of rediction errors resents an exected mean value of zero and a variance of Var e t T. The rediction errors are given by e T T ˆ T j T T j and the limits of control charts of rediction errors are defined by exressions that are resented in Table 5, where Var e T e. Table 5. Control Limits for Phase II of control charts based on rediction errors Chart UCL CL LCL e ( residuals) 3 e 3 e e (mean) A e A e R (range) D e d e D e ( standard deviation) B 6 e c 4 e B 5 e MR (moving range) D e d e D e 4. ARIMA Methodology of Box and Jenkins The aim of the methodology of Box and Jenkins is to determine and adjust the better ARIMA mathematical model to the collected rocess observations, in order to eliminate the existing autocorrelation and obtain a good rediction for each observation. To model a rocess by using this methodology, it is necessary to determine the ARIMA (, d, q) model that best fits the data, comaring the estimated autocorrelation function (EACF) with the theoretical autocorrelation function (ACF) and the estimated artial autocorrelation function (EPACF) with the theoretical artial autocorrelation function (PACF). The modeling based on the ARIMA methodology follows certain stes, namely: identification, estimation, verification, rediction. In stationary rocesses, ACF and PACF of rocess AR(), MA(q) and ARMA(,q), have distinct characteristics. Table 6 gives the different succession series. 569

6 Table 6. uccession series Process ACF PACF AR() Exonential decrease from a certain lag Presents significant eaks through lag () order which indicate the order of the model MA(q) Presents significant eaks through lag (q) Exonential decrease from a certain lag which indicate the order of the model order ARMA(,q) Exonentially decrease from a certain order lag, ositively or negatively, or switching between them Thereafter, the identified model arameters are estimated: the arameters if the behavior of the rocess is autoregressive; the arameters for moving average and variance of the error. When a satisfactory model is obtained, it is ossible to redict the otential future values of the characteristic under study and then determine the forecasting errors. 5. tudy Case The resented study case includes the study of the quality characteristic "Total Thickness", and corresonds to the aint industry rocess of a Portuguese automotive manufacturer. This study aimed the rocess control, considered of great relevance for the comany. The data collected during roduction showed the existence of significant autocorrelation, and two different aroaches have been develoed and their erformance comared. The first aroach consisted in the imlementation of the methodology suggested by the authors, using the rocess modeling and alication of hewhart charts of residuals. The second aroach was to aly the traditional charts directly to the collected data without the concern of studying a ossible significant autocorrelation. 5.. Verification of data autocorrelation To study the data autocorrelation, 93 observations related to Total Thickness are collected. The verification of autocorrelation is erformed using the "tatistica" software, calculating the EACF and EPACF for the 93 collected observations. The analysis of Figure reveals the existence of significant autocorrelation of data, since the coefficient of correlation estimated for the lag (.595) does not belong to its confidence interval. Comaring the EACF and EPACF with the ACF and PACF described in Table 6, it was found that the rocess can be modeled using a standard AR(). Lag Autocorrelation Function Total Thickness (tandard errors are white-noise estimates) Corr..E , -,5,,5, Q Conf. Limit Lag Partial Autocorrelation Function Total Thickness (tandard errors assume AR order of k-) Corr..E , -,5,,5, Conf. Limit Figure. FACE and FACPE for Total Thickness characteristic To estimate the arameters of the model AR(), the software "tatistica" is used, obtaining the values ˆ and ˆ After the rocess modeling, the residuals indeendence. 57

7 was checked with EACF and EPACF of residuals, resented in Figure. The indeendence of the residuals is found. Lag Autocorrelation Function Total Thickness: ARIMA (,,) residuals; (tandard errors are white-noise estimates) Corr..E. Q Lag Partial Autocorrelation Function Total Thickness: ARIMA (,,) residuals; (tandard errors assume AR order of k-) Corr..E , -,5,,5, Conf. Limit , -,5,,5, Conf. Limit Figure. FACE and FACPE for residuals of Total Thickness characteristic 5.. PC Alication (Phase I) 5... Control charts alied to residuals For the hewhart control charts with autocorrelated data, designed to be alied when the observations violate the assumtion of indeendence, the aroriate methodology is the following: with the 93 values used for the autocorrelation study, the control charts alied to residuals are drawn (Figure 3). During the construction of the charts, it was noticed that oint nr. was a secial cause of variation, since it exceeded the value of the UCL chart MR; the resonsibility of this situation ( MR LCMR ) lays on the value 9, reason why this value was relaced by its exected value ( ˆ ), the ARIMA model is then fitted and the revised charts of residuals are built (Figure 3). By examining this figure it is concluded that there are no secial causes of variation and the rocess is under statistical control. It is resented in Table 7 the new values of model arameters. Table 7. Parameters of the model AR () for Total Thickness characteristic Model (,,) M Residual =.486 Parameter tandard Confidence Interval 95% t Deviation -value Lower Limit Uer Limit Constant The Normality of the residuals is checked using the Kolmogorov-mirnov test ( d ; D crítico. 886 N. 99 to 5 % ; d Dcrit ). It is resented in Figure 4 the histogram of 93 values with the Kolmogorov-mirnov value, and the -value of Chi-quare test. Checked the stability of the rocess, the arameters of Total Thickness characteristic are estimated, based on the model AR(), which are resented in Table 8. 57

8 and Moving R Chart; variable: Residuals : -.7E-3 (-.7E-3); igma:.5473 (.5473); n:, Moving R:.7459 (.7459); igma:.39 (.39); n:, E Figure 3. Control chart e-mr of revised residuals of the characteristic Total Thickness 35 Variable: Residuals, Distribution: Normal Kolmogorov-mirnov d =,7447, = n.s., Lilliefors = n.s. Chi-quare test = 3,97878, df = 3 (adjusted), =, No. of observations 5 5-4,5-3,875 -,5 -,65,,65,5 3,875 4,5 Category (uer limits) Parameter ˆ Figure 4. Check of the Normality of residuals of Total Thickness characteristic Table 8. Parameters for Thickness Total characteristic Model Control Chart Process ˆ ˆ ˆ ˆ ˆ Estimate After the arameters estimation it is ossible to study the rocess caability. Once the secification is unilateral ( LL 7 ), the study of caability is carried out only based on the C (. ). It is ossible to conclude that the rocess is able to roduce at the required k C k technical secification. 57

9 5... Control chart for data without modelation When the hewhart control charts are constructed directly from the original autocorrelated data, the limits of traditional control, increases the chance of false alarms. This is illustrated in Figure 5, where it is resented the -MR control charts for 93 data of Total Thickness characteristic. The analysis of these charts indicates erroneously the existence of many secial causes of variation in the rocess (in chart, the observations 3, 8, 9, 43, 44, of whom only in instant 9 a secial cause of variation exists and the remainder are false alarms). 88 and Moving R Chart; variable: Total Thickness : 8.38 (8.38); igma:.47 (.47); n:, Moving R:.473 (.473); igma:.63 (.63); n:, Figure 5. Control chart -MR of original data of the characteristic Total Thickness 5... Comarison of the study with and without modelation Comaring both methods, and analyzing Figure 3 (residuals chart after modelation) and Figure 5 (charts alied to original data without modelation), it is quite severe not to consider the existing rocess autocorrelation. The indication given with the alication of these statistical techniques (Figure 5) may incur in severe analysis errors which are dangers for the comany, because it will imure resources to investigate some anomalies, when the rocess is actually stabilized, without secial causes of variation. 6. Conclusions The develoment of a statistical control methodology aroriated to the existence of significant rocess autocorrelation, reveals a crucial imortance, since it avoids ossible analysis errors. If the autocorrelation is not considered, some mistakes may occur, namely: ) consider a stable rocess, when several secial causes of variation are resent, which corresonds to a non stable rocess; ) consider a rocess out of statistical control, when it is actually stable (increase of false alarms); 3) incorrect estimate of the rocess arameters; 4) roceed to rough and/or incorrect analysis of the rocess caability; 5) loss of resources by a unnecessary intervention on the roductive rocess, in order to solve roblems which are not real (false alarms). 573

10 Bibliograhy Alwan, L.; Roberts, H. (988). Time-eries Modeling for tatistical Process Control. Journal of Business & Economic tatistics, Vol. 6, Gilbert, K. C.; Kirby, K.; Hild, C. R. (997). Charting Autocorrelated Data: Guidelines for Practitioners. Quality Engineering, Vol. 9, Harris, T. J.; Ross, W. H. (99). tatistical Process Control Procedures for Correlated Observations. Canadian Journal of Chemical Engineering, Vol. 69, Junior, F. J.; iedel, E. J.; Loes, L. F. (8). Comaração entre métodos utilizados no tratamento de dados autocorrelacionados no controle estatístico do rocesso. VIII encontro nacional de engenharia de rodução. Rio de Janeiro, Brasil. Maragah, H. D.; Woodall, H. W. (99). The effect of autocorrelation on the retrosective - chart. Journal of tatistical Comutation and imulation, Vol. 4,.9 4. Mastrangelo, C. M.; Montgomery, D. C. (995). PC with Correlated Observations for the Chemical and Process Industries. Quality and Reliability Engineering International, Vol., Montgomery, D. (8b). Introduction to tatistical Quality Control. 6 th ed.. New York / Jonh Wiley & ons. Montgomery, D. C.; Mastrangelo, C. M. (99). ome tatistical Process Control Methods for Autocorrelated Data. Journal of Quality Technology, Vol. 3, Pereira, Z. L.; Requeijo, J. G. (8). Qualidade: Planeamento e Controlo Estatístico de Processos. Lisboa / Prefácio. Quesenberry, C. P. (997). Methods for Quality Imrovement. New York / John Wiley & ons. Vander Weil,. A. (996). Monitoring Processes that Wander Using Integrated Moving Average Models. Technometrics, Vol. 38, Wardell, D. G.; Moskowitz, H.; Plante, R. D. (99). Control Charts in the Presence of Data Correlation. Management cience, Vol. 38, Wieringa, J. E. (999). tatistical Process Control for erially Correlated Data. Ph.D. Thesis University of Groningen. Woodall, W. H. (). Controversies and Contradictions in tatistical Process Control. Journal of Quality Technology, Vol. 3, Zhang, N. F. (998). A tatistical Control Chart for tationary Process Data. Technometrics, Vol. 4,

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